A really Simple Approximation of Smallest Grammar

In this paper we present a really simple linear-time algorithm constructing a contextfree grammar of size O(g log(N/g)) for the input string, where N is the size of the input string and g the size of the optimal grammar generating this string. The algorithm works for arbitrary size alphabets, but the running time is linear assuming that the alphabet Σ of the input string can be identified with numbers from {1, . . . , N} for some constant c. Algorithms with such an approximation guarantee and running time are known, however all of them were non-trivial and their analyses were involved. The here presented algorithm computes the LZ77 factorisation and transforms it in phases to a grammar. In each phase it maintains an LZ77-like factorisation of the word with at most ` factors as well as additional O(`) letters, where ` was the size of the original LZ77 factorisation. In one phase in a greedy way (by a left-to-right sweep and a help of the factorisation) we choose a set of pairs of consecutive letters to be replaced with new symbols, i.e. nonterminals of the constructed grammar. We choose at least 2/3 of the letters in the word and there are O(`) many different pairs among them. Hence there are O(log N) phases, each of them introduces O(`) nonterminals to a grammar. A more precise analysis yields a bound O(` log(N/`)). As ` ≤ g, this yields the desired bound O(g log(N/g)).